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Natural Selection

Regular fractals of the kind we are talking about are the outcome of iteration. Each step of the cell cycle is iterative and when the cell divides, new informa- figure 6.24 Oak leaf outline, produced by iterating a formula of less than 50 symbols. From R. J. Bird and F. Hoyle, On the Shapes of Leaves, Journal of Morphology 219 (1994) 225-241. figure 6.2s Holly leaf outline, produced by iterating a formula of less than 150 symbols. From R. J. Bird and F. Hoyle, On the Shapes of Leaves, Journal...

Life, it is often said, is just one damned thing after another. It would be hard to think of a better description of iteration. Iteration is a ubiquitous phenomenon and the operations of the world are for the most part, if not entirely, iterative. Rivers flow day after day in almost (but not quite) the same channels. Winds blow in apparently repetitive seasonal patterns planets rotate, revolve, and wobble in their orbits some clocks have hands, which rotate too. Living things reproduce...

Let us call the Euclidean space that just contains a fractal its embedding space. This space has an integral whole number of dimensions. For instance, the Koch curve can be wholly contained in two-dimensional space, and the Menger sponge in three-dimensional space. The definition that has been arrived at for fractals is that they have a dimensionality that describes how much of the embedding space they fill this is known as the fractal dimension . This accords with the use of the term dimension...

We can now propose an answer to the question of why and how mathematics works. It works because the iterative processes of the world correspond to the iterative processes of mathematics. Sometimes these iterations are explicit, as in the case of counting, multiplying, or forecasting the orbits of variables in phase space. Sometimes they are implicit, as when we use differential operators to model the gradient of an electromagnetic field. In the first type of case the correspondence is obvious...

If there is a mathematics of living things, then the most likely candidate is the mathematics of fractals. Why this is so becomes clear when we consider how fractals are created. Regular fractal patterns are made by repeatedly substituting scaled down subsections of the original pattern. As we saw in chapter 1, figure 6.2 Minaret cauliflower at three successive magnifications a-c . Reprinted from F. Grey and J. K. Kjems, Aggregates Broccoli and Cauliflower, Physica D 38 1989 154-159, with...

When all that has really been examined is some of the habits of living things, not the nature of life. Because the effects of life may be orderly it is a mistake to suppose that the process of life is itself orderly. Life, as I shall try to explain, is moderately disorderly and grows more so with the passage of time. Let us look again at what is happening in the examples we discussed. In the first, the two liquids, milk and water, are not particularly organized to begin with. Because one milk...

Chaotic systems have fractal strange attractors,5 which have a fractional dimension. As we saw in chapter 4, we can measure the fractal dimension as an indication of how chaotic the system is. Since there is a strange attractor underlying a chaotic system, we can say that chaos also has a dimension, called the correlation dimension, which is that of the fractal attractor associated with the chaotic system. The correlation dimension of a chaotic system is one of its most important features...

Far from being random, as common usage implies, chaotic systems follow strict laws that is why the functioning of such systems is called deterministic chaos. Chaotic systems are those that are influenced by very small changes in their controlling factors, like the weather, economics, social and management structures, and the brain and central nervous system. What these disparate things have in common is that they are often delicately poised between one state and another and so can easily be...